BLOCK DESIGNS WITH FACTORIAL STRUCTURE

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1 BLOCK DESIGNS WIH FACORIAL SRUCURE PRIYA KOHLI M.Sc. (Agrcultural Statstcs, Roll No. I.A.S.R.I, Lbrary Aenue, New Delh- Charperson: Dr. R. Srastaa Abstract: In block desgns wth factoral structure, there s one to one correspondence between the treatments of the block desgn and the treatment combnatons of a factoral eperment. Such a correspondence helps n analyss of the block desgns as confounded factorals. he calculus for factoral arrangements can usefully be employed to analyse desgns wth factoral eperments usng ncomplete block desgns. A block desgn s sad to hae an orthogonal factoral structure (OFS f the adjusted sum of squares due to the treatments n the block desgn can be parttoned orthogonally nto sum of squares correspondng to man effects and nteractons n a factoral eperment. Further, a complete balance s acheed oer an nteracton f and only f all the normalzed treatment contrasts belongng to a gen nteracton are estmated wth same arance. If block desgn s hang OFS and complete balance s acheed oer all nteracton components, then the desgn s sad to possess the property of OFS wth balance. Key words: Block Desgn, Factoral Eperment, Kronecker Product, Etended Group Dsble Desgn (EGD, Orthogonal Factoral Structure (OFS.. Introducton A one to one correspondence ests between the treatments of the block desgn and the treatment combnatons of a factoral eperment n a block desgn wth factoral structure. Such a correspondence helps n analyzng the block desgns as confounded factorals. A block desgn wth factoral structure requres only that man effects and nteracton contrasts estmates be adjusted for block effects. Although computer softwares, such as Statstcal Analyss System (SAS or Statstcal Package for Factoral Eperments (SPFE., can handle the analyss of block desgns wth factoral structure, the possblty of estmaton of orthogonal factoral eperments among themseles are much easer to nterpret and are, therefore preferred. We now descrbe an eample to clarfy the dea of correspondence between the block desgns and factoral eperments. Eample.: Consder a resolable Balanced Incomplete Block (BIB desgn wth parameters, b, r, k, λ and block contents as: B B B B B B Let us assocate wth ths desgn a factoral eperment wth two factors as F and F. Let the two leels of each of the two factors be denoted by and. Assgn the four treatment

2 Block Desgns wth Factoral Structure combnatons of the factoral eperment to the four treatments of the BIB desgn through the followng correspondence: ; ; ;. hs ges the followng desgn for factoral eperment: B B B B B B For the BIB desgn, the concurrence matr s of the form N N M R NK N NN rk NN (. he man effects F and F and ther nteracton F F can be estmated by usng the followng three mutually orthogonal obseratonal contrasts: able.: Contrasts for man effects and nteractons ( ( ( ( F - - F - - F F - - where (, (, (, ( denote, respectely, the total of the obseratons from the epermental unts receng the treatment combnatons,, and. he ector of coeffcents s, s and s, correspondng to these contrasts, can be wrtten as s ( (- s ( ( s ( ( ( (. where represents Kronecker product. It can be erfed that s, s and s are egen ectors of NN and hence of M. he egen alues of M are each equal to, t follows that the man effects and the two-factor s s s

3 Block Desgns wth Factoral Structure nteracton are estmated wth the same loss of nformaton. Eery BIB desgn wth m m... m n treatments can always be used as an m m... m n factoral eperment and for such desgn eery effect s estmated wth same relate loss of ( r λ nformaton. As a factoral eperment, ths class of desgns has lttle utlty rk because all the effects are confounded wth the blocks to the same etent. In a factoral eperment, one s nterested n estmatng certan effects wth full effcency. Snce there s an equal loss of nformaton on all the factoral effects, therefore, BIB desgns are of lttle use as a block desgn for factoral eperments. Certan class of Partally Balanced Incomplete Block (PBIB desgns may be useful for these eperments. Remark.: he structure of the contrasts as defned n (. reeals that the hgher order contrasts can be epressed as the Kronecker product of lower dmensonal contrasts. hs property, whch s true n general, s etremely useful n determnng the pattern of analyss. Eample.: Consder now the followng group dsble (GD desgn wth groups (, and (, and the parameters, b, r, k, λ, λ, m, n. he plan of the desgn can be obtaned by deletng the frst two blocks of the BIB desgn gen n Eample.. Blocks he concurrence matr of ths desgn s NN, (. NN NN and M. rk Here s s an egen ector of NN correspondng to the egen alue and s and s are the egen ectors of NN correspondng to the egen alue, where s, s and s are as defned n equaton (.. hus, the block desgn prodes a factoral eperment n whch the man effect F s unconfounded and s, therefore, estmated wth full effcency, whereas the man effect F and the nteracton F F are each estmated wth relate loss of nformaton. As a desgn for a factoral eperment, ths s superor to BIB desgn as ths desgn not only estmates F free from block effects but also requres less number of epermental unts. For more detaled dscusson on relate effcency of dfferent factoral

4 Block Desgns wth Factoral Structure effects n a block desgn wth factoral structure a reference may be made to Ngam et al. (988 and Gupta and Mukerjee (989.. Elements and Operatons of the Calculus (Kurkjan and Zelen, 9, 9 Consder a factoral eperment nolng n factors F, F,,F n. Let the th factor has m leels coded as,,,m ( n. A typcal selecton of leels j ( j, j,..., jn, ( j, wll be termed as the j th m, ( n treatment combnaton and the effect due to ths treatment combnaton wll be denoted as τ (j,,j n. he total number of n m treatment combnatons s. Let a (,, m - be the ector whose elements denote the leels at whch the factor F appears. hen the symbolc drect product (SDP of any two a s s defned as (a p a q (,,, m q -,,,,m q -, m p -,,m p -m q -. he treatment combnatons wll be consdered as gen by SDP among all a s,.e., a a a n. Let τ be a ector, wth elements gen by τ (j, j,...,, j n s. For eample, f n, m, m, then a, a (, a a (,,,,, and τ (τ(,, τ(,, τ(,, τ(,, τ(,, τ(,. Let Ω be the set of all n-component non-null bnary ectors. It s easy to see that there s a one-to one-correspondence between Ω and the set of all nteractons, n the sense that a typcal nteracton F F corresponds to the element (,,, n of Ω such that... F g and u for u... g,,, g. hus the n - factoral effects may be denoted by F, Є Ω. For eample, f n, then the man effects F, F and the -factor nteracton F F may be denoted by F, F and F respectely. he treatment contrasts belongng to the factoral effects may conenently be represented by makng use of Kronecker products. For each Ω, let M M n M I m, m J J ι f f (.,...(. where I s an dentty matr and J s a matr of all s both of order m m. Lemma.: (Gupta and Mukerjee, 989. For each Ω, the elements of M τ represent a complete set of treatment contrasts belongng to F. Lemma.: (Gupta and Mukerjee, 989. reatment contrasts belongng to any two dstnct nteractons are mutually orthogonal.e. M M y, for eery (,,, n and y (y, y,,y n Ω, y. Another equalent representaton n terms of orthonormal contrasts s often helpful. For n, let be the m ector wth all elements unty and P be an m m matr (

5 Block Desgns wth Factoral Structure such that the m matr ( m /, P s orthogonal. For eample, f n, m, m, then P For each m, P Ω, let n P n P...(. P P m / f f From (. and (. the relaton and (., for eery Ω,, ( n...(. P P M, whether equals or. Hence by (. P P M...(. So, analogously to M M y, t may be seen that for each, y Ω y, P P I, P P y...(. Lemma.: (Gupta and Mukerjee, 989. For each Ω, the elements of P τ represent a complete set of orthonormal contrasts belongng to the nteracton F. Now n the sequel we show that the analyss usng block desgns followng contrast analyss, Yates algorthm and able method of analyss of factoral eperments ge dentcal results. Eample.: Consder a factoral eperment conducted n a randomzed complete block desgn wth r replcatons. he treatment combnatons are gen by SDP as ( (. hese combnatons can be represented as :, :, :, :, :, :. Let the ector of treatment effects be denoted as τ ( τ, τ,..., τ... Analyss Usng Block Desgns he reduced normal equatons for estmatng the lnear functons of treatment effects are Cτ Q, wherec r[ I ], Q [ G]. A generalzed nerse of C s I. For a r set of treatment contrasts P τ such that P contans t (< lnearly ndependent rows, the - best lnear unbased estmator of P τ s gen as P τˆ P C Q and the sum of squares s (P (P C τˆ P (P τˆ. he null hypothess, H : P τ aganst the alternate hypothess ( P τˆ ( P C P ( P τˆ H : P τ can be tested usng F statstc gen by F F t, error d.f., t * MSE

6 Block Desgns wth Factoral Structure where MSE s the error mean square. For detaled descrpton on wrtng the contrasts for factoral effects, one may refer to Gupta and Mukerjee (989 and Parsad et al. (. able.: Contrasts of possble treatment combnatons of Eample. ( ( ( ( ( ( A B A*B Usng the aboe, the sum of squares due to dfferent factoral effects are as follows: Man effect A r ˆ τ P G / G / G / G / G / G / r Sum of Squares due to man effect A, SSA r ˆ ( ( ˆ ( P τ P P C τ P Man effect B P, P τ ˆ Sum of Squares due to man effect B, SSB (P (P (P τˆ P C τˆ r r

7 Block Desgns wth Factoral Structure Interacton effect AB P, P τˆ Sum of Square due to nteracton AB, SSAB(P (P (P τˆ P C τˆ r r. Yates Algorthm Yates (97 has defned a rule for calculatng factoral effects n n factoral eperments. It was later etended to the case of analyss of asymmetrcal factoral eperments. he algorthm for factoral s dscussed n the sequel. Step : Wrte the treatment combnatons n the lecographc order,.e., frst ary the leels of the frst factor from to by keepng fed the leels of second factor at. hen ary the leels of the second factor from to leel and then to leels n each of the frst treatment combnatons, so as to get treatment combnatons. Wrte these treatment combnatons n the frst column and n the second column wrte the correspondng treatment totals. Step : Dde the obseratons n the second column n groups such that each group has two obseratons, and frst half of the thrd column s flled wth the sum of the obseratons n these groups and second half wth the dfferences of the second obseraton and the frst obseraton n each group. For the factor at three leels, make the groups of three obseratons each, and n the one thrd of the net column s flled wth the sum of obseratons n these groups, net one thrd by usng the lnear component, say, -,,,.e., by takng the dfference of the thrd obseraton and the frst obseraton n each group and the rest one thrd s flled by usng the quadratc component, -,,.e., by addng the frst and thrd obseraton n each group and subtractng the twce of the second obseraton from ths sum. hese entres ge the man effect and nteractons. 7

8 Block Desgns wth Factoral Structure able.: Analyss of Varance of Factoral by Yates Algorthm reatment Step I Step II Factoral Effect Combnaton ( G ( A- - - ( - - B- - ( AB - - ( B - - ( AB where G s the Grand total. Sum of squares of man effect A, r SSA Sum of squares of man effect B, ( SSB r r ( ( Sum of squares of nteracton effect AB, ( SSAB r r r ( ( r r ( (. able Method For obtanng the sum of squares due to man effects and two factor nteractons a two-way table for factors A and B wth leels a, a and b, b, b respectely s shown below: able.: Cell totals of A and B factors a a b B b B b B A A G 8

9 Block Desgns wth Factoral Structure SSA ( A A G r r r ( ( B B B G SSB r r r ( SSAB -SSA-SSB r r ( ( r herefore, one can see that the analyss of factoral eperments conducted n block desgns does not requre any etra sklls. It only requres the knowledge of analyss of general block desgns followed by contrast analyss. he results obtaned from the other prealent methods z. able method and Yates algorthm are same. hus we can analyze factoral arrangements conenently usng block desgns followed by contrast analyss.. Orthogonal Factoral Structure (OFS and Balance In the analyss of a desgn for factoral arrangement, the epermenter s prmarly nterested n drawng conclusons on the contrasts belongng to the dfferent nteractons. A great smplfcaton occurs n nterpretng the results of analyss f the desgn has orthogonal factoral structure (OFS as descrbed n the sequel:. Orthogonal Factoral Structure A desgn for factoral eperment wll be sad to hae the orthogonal factoral structure (OFS f the Best Lnear Unbased Estmator s (BLUE s of estmable treatment contrasts belongng to dstnct nteractons are mutually orthogonal,.e., uncorrelated. When ths s realzed, n the connected case, the adjusted treatment sum of squares (SS can be splt up orthogonally nto components due to dfferent nteractons and, as such, these components may be shown n the same analyss of arance (ANOVA table. A block desgn s sad to hae an orthogonal factoral structure f the adjusted sum of squares due to treatments n the block desgn can be parttoned orthogonally nto sum of squares correspondng to man-effects and nteractons n a factoral eperment. herefore, a block desgn wth factoral structure ensures between nteractons orthogonalty. Another mportant and useful concept n contet of factoral eperments s that of balance whch can be defned as follows: Defnton.: (Shah, 98. A block desgn for a factoral eperment s called balanced f the followng condtons are satsfed: ( the desgn s equreplcate and proper; ( estmates of contrasts belongng to dfferent nteractons are uncorrelated wth each other; 9

10 Block Desgns wth Factoral Structure ( complete balance s acheed oer each of the nteractons. Complete balance s acheed oer an nteracton f and only f all the normalzed contrasts belongng to the gen nteracton are estmated wth the same arance. OFS ensures between-nteractons orthogonalty, and balance ensures wthn-nteracton orthogonalty. herefore, f a desgn has OFS and s balanced then further smplfcatons n results of analyss are acheed. Furthermore, such desgns hae elegant propertes from effcency consderatons as well. It s, therefore of nterest to eplore the algebrac and combnatoral characterzatons for desgns whch are balanced wth OFS. hs wll be useful n the actual constructon of desgns.. Algebrac Characterzatons he desgns for factoral eperments that are balanced and hae OFS hae been termed balanced factoral eperments [see e.g. (Shah, 98, 9]. hey are also known as balanced confounded desgns accordng to the nomenclature of Nar and Rao (98. Now an algebrac characterzaton for balance wth OFS n the connected case s gen. For equreplcate and proper desgns, the suffcency part of ths result was proed by Kurkjan and Zelen (9, whle the necessty part was proed by Kshrsagar (9. Gupta (98 consdered etensons to desgns that are not necessarly equreplcate or proper. he followng defnton and lemmas wll be helpful: Let Ω* be the set of all n-component bnary ectors,.e. Ω * Ω {(,,..., }. For (,..., Ω*, let n Z Z Z I f J f ( n...(. n Defnton.: (Property A A property A f t s of the form G ν ν matr G, where m, wll be sad to hae n ( Z where h(, Ω*, h εω* are real numbers. Lemma.: (Gupta and Mukerjee, 989 For a connected block desgn for factoral eperment to be balanced wth OFS, t s necessary and suffcent that the C-matr of the desgn be of the form where, C ρ M (. ρ εω (, Ω, are real numbers. heorem.: For a connected factoral desgn to be balanced wth OFS, t s necessary and suffcent that the C-matr of the desgn has property A.

11 Block Desgns wth Factoral Structure hs theorem prodes a characterzaton for balance together wth OFS n terms of property A of the C matr n the connected case. hus a desgn wll be sad to hae property A f ts C matr has property A. heorem.: (a For a connected, equreplcate block desgn wth factoral structure to hae the property OFS wth balance, t s necessary and suffcent that the matr Nk δ N has property (A. (b For a connected, equreplcate, proper block desgn for factoral eperment to hae the OFS and balance, t s necessary and suffcent that the matr NN has property (A. Let C be of the form (. then by (., (., for eery y, z ρ( y Ω, y z, P y C P y CP y ρ y I (y (. P y P y CP z, where I (y s the dentty matr of order (m - y For connected equreplcate desgn wth property A and a common replcaton number r, t s possble to ge a smple formula for the relate effcences for arous factoral effects. he dsperson matr of P y s gen as τˆ dsp (P y τˆ { ρ y } - I (y,...(. σ where I (y s the dentty matr of order m, σ s the constant error arance. For a randomzed (complete block desgn wth the same number of replcates, the dsperson matr of P y τˆ s gen as dsp(p y τˆ σ r - I (y....(. A comparson between (. and (. shows that the effcency wth respect to factoral effects F y n ths desgn s gen by, ( y ρ( y / r, Ω ε y....(. Eample.: Consder an epermental stuaton where two factors each at leels are to be tred. he epermenter has epermental unts that can be arranged n blocks of sze each. he 9 treatment combnatons are (,, (,, (,,,,,,,,. For ths set a BIB Desgn wth parameters 9, b, r, k, λ. he block contents of the BIB desgn are: y B B B B B B B7 B8 B9 B B B

12 Block Desgns wth Factoral Structure hs ges the followng desgn for factoral eperment: able.: Desgn for factoral eperment B B B B B B B7 B8 B9 B B B For the BIB Desgn, the concurrence matr s of the form ( Z Z I I J J NN, where Z s as defned earler. ( ( Z Z Z Z Z NN I I C k r P P Z P I, P Z P, P CP I P P Z P I, P Z P, P CP I P P Z P I, P Z P, P CP I able.: he effcences for man effects and nteractons, ε, ε, ε

13 Block Desgns wth Factoral Structure hs shows that the man effects as well as the nteracton effects hae same effcency. herefore, as mentoned earler a BIB desgn for a factoral eperment may not be a good desgn and one has to look for alternate desgns that ensure estmaton of some factoral effects (possbly lower order wthout loss of nformaton. Wth ths n ew, we delete frst three blocks of the aboe BIB desgn and get Sem Regular Group Dsble (SRGD desgn wth parameters 9, b 9, r, k, λ, λ, m, n.he block contents of ths desgn are gen as: B B B B B B B7 B8 B9 NN Z Z Z C Z Z Z P CP I, P CP I, P CP I able.: he effcences for man effects and nteractons ε ε (, ε (, (, Here the man effect A s unconfounded and has full effcency. Further f we delete last s blocks, we get a Rectangular desgn wth parameters 9, b, r, k, λ, λ, λ. he block contents are gen as B B B B B B he concurrence matr for the aboe desgn s NN Z Z C Z Z Z P CP I, P CP I, P CP I

14 Block Desgns wth Factoral Structure able.: he effcences for man effects and nteractons (, ε (, ε ε (, hese nteracton effcences n the desgn shows that the nteracton s unconfounded and has full effcency. Smlarly one can obtan the effcences of dfferent factoral effects by deletng each of the possble pars of blocks, From the aboe, t can be seen that BIB desgns are not good desgns for factoral eperments. he partally balanced ncomplete block desgns, howeer, are to some etent sutable. he most general class of partally balanced desgns.e. sutable for factoral eperments s Etended Group Dsble (EGD desgns. An EGD desgn wheneer ests possesses the property of OFS wth balance.. Etended Group Dsble (EGD Desgns Let there be ss... s treatments φ ( α, α,..., α p. wo treatments φ( α, α,..., α p and φ( β, β,..., βp are ( u, u,..., u p th assocates f { u ( α β, u( α β,..., u( α p β p } ( u, u,..., u p, where u ( s a functon of wth u ( f and u ( f. Obously, eery treatment s the (, -th assocate of tself, and of no other treatment. In ths assocaton scheme, the assocate classes are not numbered n decmal system; nstead they are epressed n bnary notaton. he number of assocate classes, m p (ecludng (, -th class. hs scheme has been studed by Hnkelmann and Kempthorne (9 and Hnkelmann (9 under the name EGD assocaton scheme. Defnton.: An arrangement of the m treatment combnatons n b blocks each of sze k wll be called an etended group dsble (EGD desgn f ( the desgn s bnary n the sense that each treatment combnaton occurs at most once n each block, ( each treatment combnaton occurs n eactly r blocks and ( eery two dstnct treatment combnatons, whch are th assocates of each other, occur together n λ( blocks, Ω. Eample.: Consder a factoral arranged n twele blocks as shown below: B B B B B B B7 B8 B9 B B B he desgn s connected, proper wth constant block sze k, and equreplcate wth common replcaton number r. hs s an EGD desgn wth parameters, n, m, m, b, r k, λ λ and λ.

15 Block Desgns wth Factoral Structure Block desgns wth factoral structure prodes a class of desgns whch smplfes the comple analyss of the factoral eperments. References Ahuja, S., Parsad, R. and Gupta, V. K. (. Statstcsl package for factoral eperments (SPFE.. IASRI, New Delh. Gupta, S. C. (98. A basc lemma and the analyss of block and kronecker product desgns. Journal of Statst. Plann. Inf. 7, 7-. Gupta, S. and Mukerjee, R. (989. Lecture Notes n Statstcs-A Calculus for Factoral Arrangements (9. Sprnger-Verlag Inc. Hnkelmann, K. (9. Etended group dsble partally balanced ncomplete block desgns. Annals of Mathematcal Statst.,, 8-9. Hnkelmann, K. and Kempthorne, O. (9. wo classes of group dsble partal dallel crosses. Bometrka,, 8-9. Kshrsagar, A. M. (9. Balanced factoral desgns. J. R. Statst. Soc. B8, 9-9. Kurkjan, B. and Zelen, M. (9. A calculus for factoral arrangements. Annals of Mathematcal Statstcs,, -9. Kurkjan, B. and Zelen, M. (9. Applcatons of the calculus for factoral arrangements. I: block and drect product desgns. Bometrka,, -7. Nar, K.R. and Rao, C.R. (98. Confoundng n asymmetrc factoral eperments. J. Roy. Statst. Soc. B,, 9-. Ngam, A. K., Pur, P. D. and Gupta, V. K. (988. Characterzatons and Analyss of Block Desgns, Wley Eastern Lmted, New Delh. Parsad, R., Gupta, V. K., Batra, P. K., Srastaa, R., Kaur, R., Kaur, A. and Arya, P. (. A dagnostc study of desgn and analyss of feld eperments. Project Report, IASRI, New Delh. Shah, B. V. (98. On balancng n factoral eperments. Annals of Mathematcal Statstcs, 9, Shah, B. V. (9a. Balanced factoral eperments. Annals of Mathematcal Statstcs,, -. Yates, F. (97. he desgn and analyss of factoral eperments. Imperal Bureau of Sol Scence. echncal Commun. No.. Some Addtonal References Das, M. N. and Gr, N. C. (98. Desgn and analyss of eperments. Second Edton. Wley Eastern: New Delh. Gupta, S. C. (98. Some new methods for constructng block desgns hang orthogonal factoral structure. J. R. Statst. Soc. B, Dey, A. (98. heory of block desgns. Wley Eastern Lmted, New Delh.